The conservation theorem for differential nets

Abstract

We prove the conservation theorem for differential nets – the graph-theoretical syntax of the differential extension of Linear Logic (Ehrhard and Regnier's DiLL). The conservation theorem states that the property of having infinite reductions (here infinite chains of cut elimination steps) is preserved by non-erasing steps. This turns the quest for strong normalisation (SN) into one for non-erasing weak normalisation (WN), and indeed we use this result to prove SN of simply typed DiLL (with promotion). Along the way to the theorem we achieve a number of additional results having their own interest, such as a standardisation theorem and a slightly modified system of nets, DiLL∂ϱ.

An introduction to differential linear logic: proof-nets, models and antiderivatives

“…It is then possible to prove basic properties such as confluence and normalization † . For these topics, we refer mainly to the work of Pagani (2009), Pagani and Tranquilli (2009), Tranquilli (2009), Pagani and Tranquilli (2011), Gimenez (2011). We also refer to Vaux (2009) for the link between the algebraic properties of k and the properties of ;, in a simpler λ-calculus setting.…”

Factorization in Call-by-Name and Call-by-Value Calculi via Linear Logic

“…Least-level reduction is studied for calculi based on linear-logic in [34,1] and for linear logic proof-nets in [8,26]. It is studied for pure CbN λ-calculus in [2].…”

Factorization in Call-by-Name and Call-by-Value Calculi via Linear Logic

“…Least-level reduction is studied for linear-logic-based calculi in [33,2] and for linear logic proof-nets in [9,26]. Least-level factorization and normalization for the pure CbN λ-calculus is studied in [3].…”